On the Semi-Local Convergence of a Fifth-Order Convergent Method for Solving Equations

We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our technique of recurrent functions. Finally, a numerical example is given.


Introduction
Let us consider an equation Here, G : Ω ⊂ X → Y is a nonlinear Fréchet-differentiable operator, X and Y are Banach spaces, Ω is an open convex subset of X. To find the approximate solution x * ∈ Ω of (1), iterative methods are used very often. The most popular is the quadratically convergent Newton method [1][2][3]. To increase the order of convergence, multi-step methods have been developed [4][5][6][7][8][9][10][11][12]. Multipoint iterative methods for solving the nonlinear equation have advantages over one-point methods because they have higher orders of convergence and computational efficiency. Furthermore, some methods need to compute only one derivative or divided difference per one iteration.
In this article, we consider the method with the fifth-order convergent x k+1 = z k − G (y k ) −1 G(z k ), k = 0, 1, . . . , where T k = G (x k ) + G (y k ). It was proposed in [9]. However, the local convergence was shown using Taylor expansions and required the existence of six-order derivatives not used on (2) in the proof of the main result. The semi-local convergence has not been studied. This is the purpose of this paper. We also only use the first derivative, which only appears in (2).
To study the multi-step method, it is often required that the operator F be a sufficiently differentiable function in a neighborhood of solutions. This restricts the applicability of methods. Let us consider the function where ϕ : Ω ⊂ R → R, Ω = [−0.5, 1.5]. This function has zero t * = 1, and ϕ (t) = 6 log t 2 + 60t 2 − 24t + 22. Obviously, ϕ (t) is not bounded on Ω. Therefore, the convergence of Method (2) is not guaranteed by the analysis in the previous paper. That is why we develop a semi-local convergence analysis of Method (2) under classical Lipschitz conditions for first-order derivatives only. Hence, we extend the applicability of the method. There is a plethora of single, two-step, three-step, and multi-step methods whose convergence has been shown using the second or higher-order derivatives or divided differences [1][2][3][5][6][7][8][9][10]12]. The paper is organized as follows: Section 2 deals with the convergence of scalar majorizing sequences. Section 3 gives the semi-local convergence analysis of Method (2) and the uniqueness of solution. The numerical example is shown in Section 4.

Majorizing Sequence
Let L 0 , L and η be positive parameters. Define scalar sequences {δ k }, {µ k } and {σ k } by Sequence (3) shall be shown to be majorizing for Method (2) in Section 3. However, first, we present some convergence results for Method (2).
Then, Sequence (3) is bounded from above by 1 L 0 , nondecreasing and lim Proof. It follows by the definition of sequence {δ k } and (4) that it is bounded from above by 1 L 0 and non-decreasing, so it converges to δ * .

It follows
Then, evidently, (A or if where recurrent polynomials are defined on the interval [0, 1) A connection between two consecutive polynomials is needed In particular, one obtains Define the function on the interval [0, 1) by Then, by (8), one has f (1) Hence, (7) holds if which is true by (5).
Similarly, instead of (A i ), one can show Then, (A or if where This time one obtains In particular, one has It follows from (16) and (18) that Therefore, (15) holds if which is true by (5).
where we also used Then, (21) holds if As in (8), one obtains In particular, one obtains that Define the function on the interval [0, 1) by In view of (24) and (27), we have Hence, (23) holds if which is true by (5).
We also used which is true since i ) is completed. It follows that sequence {δ i } is bounded from above by δ * * and in non-decreasing, and as such, it converges to δ * .

Semi-Local Convergence
The hypotheses (H) are needed. Assume:

Hypothesis 3. Restricted Lipschitz condition G
for all v, w ∈ Ω 1 and some L > 0.
The main Semi-local result for Method (2) is shown next using the hypotheses (H).
By (H1), one obtains k ) holds. Let w ∈ U(x 0 , δ * ). Then, it follows from (H1) and (H2) that follow by a Lemma on linear invertible operators due to Banach [3,13]. Notice also that x 1 is well-defined by the third substep of Method (2) for k = 0 since y 0 ∈ U(x 0 , δ * ). Next, the linear operator (G (x k ) + G (y k )) is shown to be invertible. Indeed, one obtains by (H2): In particular, z 0 is well-defined by the second substep of Method (2) for k = 0. Moreover, we can write Hence, by (31) (for w = x 0 ), (32), (33), (H2) and (3), one obtains where we also used This shows (B (1) k ) for k = 0. Moreover, one has One can write by the second substep of Method (2): so by (H3): Hence, by (3), (31) (for w = y k ), and (35), which shows (B (2) k ). Using the third subset of Method (2), one has so by (H3), (31) (for w = x k+1 ), and the induction hypotheses The following have also been used Hence, the induction for items (B k ) is completed. Moreover, because of x k , y k , z k ∈ U(x 0 , δ * ), sequence {δ k } is fundamental since X is a Banach space. Therefore, there exists x * ∈ U[x 0 , δ * ] such that lim k→∞ x k = x * . By (37), one obtains It follows that G(x k ) = 0, where the continuity of G is also used. Let j ≥ 0. Then, from the estimate one obtains (30) by letting j → ∞.
A uniqueness of the solution result follows next.

Theorem 2.
Assume: Let Then, the only solution of Equation (1) in the region Ω 2 is x * .

Conclusions
A semi-local convergence analysis of the Newton-type method that is fifth-order convergent is provided under the classical Lipschitz conditions for first-order derivatives. The regions of convergence and uniqueness of the solution are established. The results of a numerical experiment are given.